Some contributions to k-contact Lagrangian field equations, symmetries and dissipation laws

Abstract

It is well known that k-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the k-contact Euler-Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the k-contact Euler-Lagrange equations written in terms of k-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kind of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and k-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyse the relation between k-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.